Section: Scientific Foundations
Proof search in Type Theory
Cross-fertilizing ideas between the proof search approach and the
proof normalization approach, Lengrand has interacted with the TypiCal
(INRIA Saclay) and the 
r2 (INRIA Rocquencourt) project-teams.
In proof assistants based on the proof normalization approach, or Type Theory, it is a hard challenge to design and understand their proof search mechanisms. Based on ideas from [49] , a major effort has been spent on using concepts from the proof search approach, like focused proof systems , in order to rationalize the implemented mechanisms.
By doing so, we have helped improve the Coq system, by impacting the design of the new version of the tool's proof engine. One of these proof search mechanisms, known as pattern unification , has again become a hot topic of Coq's design, after Lengrand's use of Coq to specify a particular algorithm has revealed a drastic need for this missing feature.
It also emerged from Lengrand's interaction with these project-teams, that bridging Type Theory with the proof theory developed at Parsifal confirms the need for more extensionality on the functions programmed in Coq. Efforts to add such extensionality are ongoing.
